how to snap (align) edge to (with) a vertex in another object
I think it might be that you're not moving the edge under a constraint, so you can't hover over the target without dragging the whole edge down to it. 

The 'Active', 'Closest' etc. settings refer to the source of the snap, the moving element. The 'Vertex', 'Face', etc. button next door refers to the target, the still element.

The most intuitive setting, I think, is (source)'Active', (target)'Vertex', used with a selection of source vertices made, one of them active, aiming to align with a target vertex. 

So try it that way, with all the vertices of your edge(loop?) selected, using GX, constraining the move, leaving your mouse free to hover over the target vertex, without pulling the selection with it.